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Theorem supmaxlemOLD 7982
 Description: A set that contains the greatest element satisfies the antecedent in supremum theorems. This allows to be used in some situations without the completeness axiom. (Contributed by Jeff Hoffman, 17-Jun-2008.) Obsolete as of 30-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
supmaxlemOLD
Distinct variable groups:   ,   ,,,   ,,,   ,,,
Allowed substitution hints:   (,)

Proof of Theorem supmaxlemOLD
StepHypRef Expression
1 breq2 4406 . . . . . . 7
21rspcev 3150 . . . . . 6
32ex 436 . . . . 5
43ralrimivw 2803 . . . 4
5 breq2 4406 . . . . . . 7
65notbid 296 . . . . . 6
76cbvralv 3019 . . . . 5
87biimpi 198 . . . 4
94, 8anim12ci 571 . . 3
10 breq1 4405 . . . . . . 7
1110notbid 296 . . . . . 6
1211ralbidv 2827 . . . . 5
13 breq2 4406 . . . . . . 7
1413imbi1d 319 . . . . . 6
1514ralbidv 2827 . . . . 5
1612, 15anbi12d 717 . . . 4
1716rspcev 3150 . . 3
189, 17sylan2 477 . 2
19183impb 1204 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 371   w3a 985   wceq 1444   wcel 1887  wral 2737  wrex 2738   class class class wbr 4402 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403 This theorem is referenced by:  supmaxOLD  7983
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